Explicit images for the Shimura Correspondence

TL;DR

This paper provides explicit formulas and constructive proofs for the Shimura correspondence, extending previous results by Yang, for certain modular forms and multipliers, including eta-multiplier and theta-quotients.

Contribution

It offers explicit formulas and a constructive proof for the Shimura lift associated with eta-multiplier, expanding on Yang's earlier trace-based proofs.

Findings

Explicit formulas for the Shimura lift for odd r between 1 and 23

Constructive proof of Yang's result for eta-multiplier

Formulas for lifts of Hecke eigenforms and Rankin-Cohen brackets

Abstract

In 2014, Yang showed that for $F \in \mathcal{A}_{r, s, 1, 1_N}$, we have $\textup{Sh}_{r}(F \mid V_{24}) = G \otimes \chi_{12}$ where $G\in S^{new}_{r+2s - 1}(\Gamma_{0}(6), - \left( \frac{8}{r} \right), - \left( \frac{12}{r} \right))$, where $\textup{Sh}_{r}$ is the $r$-th Shimura lift associated to the theta-multiplier. He proved a similar result for $(r,6) = 3$.\:His proofs rely on trace computations in integral and half-integral weights. In this paper, we provide a constructive proof of Yang's result. We obtain explicit formulas for $\mathcal{S}_{r}(F)$, the $r$-th Shimura lift associated to the eta-multiplier defined by Ahlgren, Andersen, and Dicks, when $1\leq r\leq 23$ is odd and $N = 1$. We also obtain formulas for lifts of Hecke eigenforms multiplied by theta-function eta-quotients and lifts of Rankin-Cohen brackets of Hecke eigenforms with theta-function eta-quotients.